Question

expand the expression to a polynomial in standard form:
$(2x^2 - 3x - 3)(2x^2 + 3x + 2)$

Explanation:

Step1: Use distributive property (FOIL-like for trinomials)

Multiply each term in the first trinomial by each term in the second trinomial:
\[

$$\begin{align*} &(2x^2 - 3x - 3)(2x^2 + 3x + 2)\\ =&2x^2\cdot2x^2 + 2x^2\cdot3x + 2x^2\cdot2 - 3x\cdot2x^2 - 3x\cdot3x - 3x\cdot2 - 3\cdot2x^2 - 3\cdot3x - 3\cdot2 \end{align*}$$

\]

Step2: Simplify each product

Calculate each term:
\[

$$\begin{align*} &4x^4 + 6x^3 + 4x^2 - 6x^3 - 9x^2 - 6x - 6x^2 - 9x - 6 \end{align*}$$

\]

Step3: Combine like terms

Combine terms with the same power of \(x\):

• For \(x^4\): \(4x^4\)
• For \(x^3\): \(6x^3 - 6x^3 = 0\)
• For \(x^2\): \(4x^2 - 9x^2 - 6x^2 = -11x^2\)
• For \(x\): \(-6x - 9x = -15x\)
• Constant term: \(-6\)

So the simplified polynomial is \(4x^4 - 11x^2 - 15x - 6\)

Answer:

\(4x^4 - 11x^2 - 15x - 6\)